Journal of Modern Dynamics (JMD)

Algebraically periodic translation surfaces

Pages: 209 - 248, Issue 2, April 2008      doi:10.3934/jmd.2008.2.209

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Kariane Calta - Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, United States (email)
John Smillie - Department of Mathematics, Cornell University, Ithaca, NY 14853, United States (email)

Abstract: We develop an algebraic framework for studying translation surfaces. We study the Sah--Arnoux--Fathi-invariant and the collection of directions in which it vanishes. We show that these directions are described by a number field which we call the periodic direction field. We study the $J$-invariant of a translation surface, introduced by Kenyon and Smillie and used by Calta. We relate the $J$-invariant to the periodic direction field. For every number field $K\subset\ \mathbb R$ we show that there is a translation surface for which the periodic direction field is $K$. We study automorphism groups associated to a translation surface and relate them to the $J$-invariant. We relate the existence of decompositions of translation surfaces into squares with the total reality of the periodic direction field.

Keywords:  translation surfaces, algebraic periodicity, J-invariant, SAF-invariant.
Mathematics Subject Classification:  Primary: 57M50, Secondary: 37D50, 30F30.

Received: May 2007;      Revised: October 2007;      Available Online: January 2008.